?DispersionA beam of white light goes from air into water at an incident angle of

The Propagation of Light Under what conditions can light be modeled like a ray? Like a wave?

The Propagation of Light Why is the index of refraction always greater than or equal to 1?

The Propagation of Light Does the fact that the light flash from lightning reaches you before its sound prove that the speed of light is extremely large or simply that it is greater than the speed of sound? Discuss how you could use this effect to get an estimate of the speed of light.

The Propagation of Light Speculate as to what physical process might be responsible for light traveling more slowly in a medium than in a vacuum.

The Law of Reflection Using the law of reflection, explain how powder takes the shine off of a person’s nose. What is the name of the optical effect?

Refraction Diffusion by reflection from a rough surface is described in this chapter. Light can also be diffused by refraction. Describe how this occurs in a specific situation, such as light interacting with crushed ice.

Refraction Will light change direction toward or away from the perpendicular when it goes from air to water? Water to glass? Glass to air?

Refraction Explain why an object in water always appears to be at a depth shallower than it actually is?

Refraction Explain why a person’s legs appear very short when wading in a pool. Justify your explanation with a ray diagram showing the path of rays from the feet to the eye of an observer who is out of the water.

Refraction Explain why an oar that is partially submerged in water appears bent.

Total Internal Reflection A ring with a colorless gemstone is dropped into water. The gemstone becomes invisible when submerged. Can it be a diamond? Explain.

Total Internal Reflection The most common type of mirage is an illusion that light from faraway objects is reflected by a pool of water that is not really there. Mirages are generally observed in deserts, when there is a hot layer of air near the ground. Given that the refractive index of air is lower for air at higher temperatures, explain how mirages can be formed.

The Propagation of Light What is the speed of light in water? In glycerine?

The Propagation of Light What is the speed of light in air? In crown glass?

The Propagation of Light Calculate the index of refraction for a medium in which the speed of light is \(2.012 \times 10^{8} \mathrm{~m} / \mathrm{s}\), and identify the most likely substance based on Table 1.1 . Text Transcription: 2.012 times 10^8 m/s

The Propagation of Light In what substance in Table 1.1 light is the speed of \(2.290 \times 10^{8} \mathrm{~m} / \mathrm{s} ?\)? Text Transcription: 2.290 times 10^8 m/s

The Propagation of Light There was a major collision of an asteroid with the Moon in medieval times. It was described by monks at Canterbury Cathedral in England as a red glow on and around the Moon. How long after the asteroid hit the Moon, which is \(3.84 \times 10^{5} \mathrm{~km}\) away, would the light first arrive on Earth? Text Transcription: 3.84 times 10^5 km

The Propagation of Light Components of some computers communicate with each other through optical fibers having an index of refraction n = 1.55. What time in nanoseconds is required for a signal to travel 0.200 m through such a fiber?

The Propagation of Light Compare the time it takes for light to travel 1000 m on the surface of Earth and in outer space.

The Propagation of Light How far does light travel underwater during a time interval of \(1.50 \times 10^{-6} s\) Text Transcription: 1.50 times 10^-6 s

The Law of Reflection Suppose a man stands in front of a mirror as shown below. His eyes are 1.65 m above the f loor and the top of his head is 0.13 m higher. Find the height above the floor of the top and bottom of the smallest mirror in which he can see both the top of his head and his feet. How is this distance related to the man’s height?

The Law of Reflection Show that when light reflects from two mirrors that meet each other at a right angle, the outgoing ray is parallel to the incoming ray, as illustrated below.

The Law of Reflection On the Moon’s surface, lunar astronauts placed a corner reflector, off which a laser beam is periodically reflected. The distance to the Moon is calculated from the round-trip time. What percent correction is needed to account for the delay in time due to the slowing of light in Earth’s atmosphere? Assume the distance to the Moon is precisely \(3.84 \times 10^{8} \mathrm{~m}\) and Earth’s atmosphere (which varies in density with altitude) is equivalent to a layer 30.0 km thick with a constant index of refraction n= 1.000293. Text Transcription: 3.84 times 10^8 m

The Law of Reflection A flat mirror is neither converging nor diverging. To prove this, consider two rays originating from the same point and diverging at an angle \(theta\) (see below). Show that after striking a plane mirror, the angle between their directions remains \(theta\). Text Transcription: theta

Refraction Unless otherwise specified, for problems 1 through 10, the indices of refraction of glass and water should be taken to be 1.50 and 1.333, respectively. A light beam in air has an angle of incidence of \(35^{\circ}\) at the surface of a glass plate. What are the angles of reflection and refraction? Text Transcription: 35 degrees

Refraction Unless otherwise specified, for problems 1 through 10, the indices of refraction of glass and water should be taken to be 1.50 and 1.333, respectively. A light beam in air is incident on the surface of a pond, making an angle of \(20^{\circ}\) with respect to the surface. What are the angles of reflection and refraction? Text Transcription: 20 degrees

Refraction Unless otherwise specified, for problems 1 through 10, the indices of refraction of glass and water should be taken to be 1.50 and 1.333, respectively. When a light ray crosses from water into glass, it emerges at an angle of \(30^{\circ}\) with respect to the normal of the interface. What is its angle of incidence? Text Transcription: 30 degrees

Refraction Unless otherwise specified, for problems 1 through 10, the indices of refraction of glass and water should be taken to be 1.50 and 1.333, respectively. A pencil flashlight submerged in water sends a light beam toward the surface at an angle of incidence of \(30^{\circ}\). What is the angle of refraction in air? Text Transcription: 30 degrees

Refraction Unless otherwise specified, for problems 1 through 10, the indices of refraction of glass and water should be taken to be 1.50 and 1.333, respectively. Light rays from the Sun make a \(30^{\circ}\) angle to the vertical when seen from below the surface of a body of water. At what angle above the horizon is the Sun? Text Transcription: 30 degrees

Refraction Unless otherwise specified, for problems 1 through 10, the indices of refraction of glass and water should be taken to be 1.50 and 1.333, respectively. The path of a light beam in air goes from an angle of incidence of \(35^{\circ}\) to an angle of refraction of \(22^{\circ}\) when it enters a rectangular block of plastic. What is the index of refraction of the plastic? Text Transcription: 35 degrees 22 degrees

Refraction Unless otherwise specified, for problems 1 through 10, the indices of refraction of glass and water should be taken to be 1.50 and 1.333, respectively. A scuba diver training in a pool looks at his instructor as shown below. What angle does the ray from the instructor’s face make with the perpendicular to the water at the point where the ray enters? The angle between the ray in the water and the perpendicular to the water is \(25.0^{\circ}\). Text Transcription: 25.0 degrees

Refraction Unless otherwise specified, for problems 1 through 10, the indices of refraction of glass and water should be taken to be 1.50 and 1.333, respectively. (a) Using information in the preceding problem, find the height of the instructor’s head above the water, noting that you will first have to calculate the angle of incidence. (b) Find the apparent depth of the diver’s head below water as seen by the instructor.

Total Internal Reflection Verify that the critical angle for light going from water to air is \(48.6^{\circ}\) as discussed at the end of Example 1.4, regarding the critical angle for light traveling in a polystyrene (a type of plastic) pipe surrounded by air. Text Transcription: 48.6 degrees

Total Internal Reflection (a) At the end of Example 1.4 , it was stated that the critical angle for light going from diamond to air is \(24.4^{\circ}\). Verify this. (b) What is the critical angle for light going from zircon to air? Text Transcription: 24.4 degrees

Total Internal Reflection An optical fiber uses flint glass clad with crown glass. What is the critical angle?

Total Internal Reflection At what minimum angle will you get total internal reflection of light traveling in water and reflected from ice?

Total Internal Reflection Suppose you are using total internal reflection to make an efficient corner reflector. If there is air outside and the incident angle is \(45.0^{\circ}\), what must be the minimum index of refraction of the material from which the reflector is made? Text Transcription: 45.0 degrees

Additional Problems From his measurements, Roemer estimated that it took 22 min for light to travel a distance equal to the diameter of Earth’s orbit around the Sun. (a) Use this estimate along with the known diameter of Earth’s orbit to obtain a rough value of the speed of light. (b) Light actually takes 16.5 min to travel this distance. Use this time to calculate the speed of light.

Additional Problems Cornu performed Fizeau’s measurement of the speed of light using a wheel of diameter 4.00 cm that contained 180 teeth. The distance from the wheel to the mirror was 22.9 km. Assuming he measured the speed of light accurately, what was the angular velocity of the wheel?

Additional Problems Suppose you have an unknown clear substance immersed in water, and you wish to identify it by finding its index of refraction. You arrange to have a beam of light enter it at an angle of \(45.0^{\circ}\), and you observe the angle of refraction to be \(40.3^{\circ}\). What is the index of refraction of the substance and its likely identity? Text Transcription: 45.0 degrees 40.3 degrees

Additional Problems Shown below is a ray of light going from air through crown glass into water, such as going into a fish tank. Calculate the amount the ray is displaced by the glass \((\Delta x)\), given that the incident angle is \(40.0^{\circ}\) and the glass is 1.00 cm thick. Text Transcription: Delta x 40.0 degrees

Additional Problems Considering the previous problem, show that \(\theta_{3}\) is the same as it would be if the second medium were not present. Text Transcription: theta_3

Additional Problems At what angle is light inside crown glass completely polarized when reflected from water, as in a fish tank?

Additional Problems Light reflected at \(55.6^{\circ}\) from a window is completely polarized. What is the window’s index of refraction and the likely substance of which it is made? Text Transcription: 55.6 degrees

Challenge Problems Light shows staged with lasers use moving mirrors to swing beams and create colorful effects. Show that a light ray reflected from a mirror changes direction by \(2 \theta\) when the mirror is rotated by an angle \(\theta\) Text Transcription: 2 theta theta

Challenge Problems Consider sunlight entering Earth’s atmosphere at sunrise and sunset-that is, at a \(90.0^{\circ}\) incident angle. Taking the boundary between nearly empty space and the atmosphere to be sudden, calculate the angle of refraction for sunlight. This lengthens the time the Sun appears to be above the horizon, both at sunrise and sunset. Now construct a problem in which you determine the angle of refraction for different models of the atmosphere, such as various layers of varying density. Your instructor may wish to guide you on the level of complexity to consider and on how the index of refraction varies with air density. Text Transcription: 90.0 degrees

Challenge Problems A light ray entering an optical fiber surrounded by air is first refracted and then reflected as shown below. Show that if the fiber is made from crown glass, any incident ray will be totally internally reflected.

Challenge Problems A light ray falls on the left face of a prism (see below) at the angle of incidence \(\theta\) for which the emerging beam has an angle of refraction \(\theta\) at the right face. Show that the index of refraction n of the glass prism is given by \(n=\frac{\sin \frac{1}{2}(\alpha+\phi)}{\sin \frac{1}{2} \phi}\) where \(\phi\) is the vertex angle of the prism and \(\alpha\) is the angle through which the beam has been deviated. If \(\alpha=37.0^{\circ}\) and the base angles of the prism are each \(50.0^{\circ}\), what is n? Text Transcription: theta n=sin 1/2 (alpha+phi)/sin 1/2 phi phi alpha alpha=37.0 degrees 50.0 degrees

Total Internal Reflection How can you use total internal reflection to estimate the index of refraction of a medium?

Dispersion Is it possible that total internal reflection plays a role in rainbows? Explain in terms of indices of refraction and angles, perhaps referring to that shown below. Some of us have seen the formation of a double rainbow; is it physically possible to observe a triple rainbow? (credit: "Chad"/Flickr)

Dispersion A high-quality diamond may be quite clear and colorless, transmitting all visible wavelengths with little absorption. Explain how it can sparkle with flashes of brilliant color when illuminated by white light.

Huygens’s Principle How do wave effects depend on the size of the object with which the wave interacts? For example, why does sound bend around the corner of a building while light does not?

Huygens’s Principle Does Huygens’s principle apply to all types of waves?

Huygens’s Principle If diffraction is observed for some phenomenon, it is evidence that the phenomenon is a wave. Does the reverse hold true? That is, if diffraction is not observed, does that mean the phenomenon is not a wave?

Polarization Can a sound wave in air be polarized? Explain.

Polarization No light passes through two perfect polarizing filters with perpendicular axes. However, if a third polarizing filter is placed between the original two, some light can pass. Why is this? Under what circumstances does most of the light pass?

Polarization Explain what happens to the energy carried by light that it is dimmed by passing it through two crossed polarizing filters.

Polarization When particles scattering light are much smaller than its wavelength, the amount of scattering is proportional to \(\frac{1}{\lambda}\). Does this mean there is more scattering for small \(\lambda\) than large \(\lambda\)? How does this relate to the fact that the sky is blue? Text Transcription: 1/lambda lambda

Polarization Using the information given in the preceding question, explain why sunsets are red.

Polarization When light is reflected at Brewster’s angle from a smooth surface, it is 100% polarized parallel to the surface. Part of the light will be refracted into the surface. Describe how you would do an experiment to determine the polarization of the refracted light. What direction would you expect the polarization to have and would you expect it to be 100%.

Polarization If you lie on a beach looking at the water with your head tipped slightly sideways, your polarized sunglasses do not work very well. Why not?

Total Internal Reflection You can determine the index of refraction of a substance by determining its critical angle. (a) What is the index of refraction of a substance that has a critical angle of \(68.4^{\circ}\) when submerged in water? What is the substance, based on Table 1.1? (b) What would the critical angle be for this substance in air? Text Transcription: 68.4 degrees

Total Internal Reflection A ray of light, emitted beneath the surface of an unknown liquid with air above it, undergoes total internal reflection as shown below. What is the index of refraction for the liquid and its likely identification?

Total Internal Reflection Light rays fall normally on the vertical surface of the glass prism (n = 1.50) shown below. (a) What is the largest value for \(\phi\) such that the ray is totally reflected at the slanted face? (b) Repeat the calculation of part (a) if the prism is immersed in water.

Dispersion (a) What is the ratio of the speed of red light to violet light in diamond, based on Table 1.2? (b) What is this ratio in polystyrene? (c) Which is more dispersive?

Dispersion A beam of white light goes from air into water at an incident angle of \(75.0^{\circ}\). At what angles are the red (660 nm) and violet (410 nm) parts of the light refracted? Text Transcription: 75.0 degrees

Dispersion By how much do the critical angles for red (660 nm) and violet (410 nm) light differ in a diamond surrounded by air?

Dispersion (a) A narrow beam of light containing yellow (580 nm) and green (550 nm) wavelengths goes from polystyrene to air, striking the surface at a \(30.0^{\circ}\) incident angle. What is the angle between the colors when they emerge? (b) How far would they have to travel to be separated by 1.00 mm? Text Transcription: 30.0 degrees

Dispersion A parallel beam of light containing orange (610 nm) and violet (410 nm) wavelengths goes from fused quartz to water, striking the surface between them at a \(60.0^{\circ}\) incident angle. What is the angle between the two colors in water? Text Transcription: 60.0 degrees

Dispersion A ray of 610-nm light goes from air into fused quartz at an incident angle of \(55.0^{\circ}\) At what incident angle must 470 nm light enter flint glass to have the same angle of refraction? Text Transcription: 55.0 degrees

Dispersion A narrow beam of light containing red (660 nm) and blue (470 nm) wavelengths travels from air through a 1.00-cm-thick flat piece of crown glass and back to air again. The beam strikes at a \(30.0^{\circ}\) incident angle. (a) At what angles do the two colors emerge? (b) By what distance are the red and blue separated when they emerge? Text Transcription: 30.0 degrees

Dispersion A narrow beam of white light enters a prism made of crown glass at a \(45.0^{\circ}\) incident angle, as shown below. At what angles, \(\theta_{\mathrm{R}} \text { and } \theta_{\mathrm{V}}\), do the red (660 nm) and violet (410 nm) components of the light emerge from the prism? Text Transcription: 45.0 degrees theta_R and theta_V

Polarization What angle is needed between the direction of polarized light and the axis of a polarizing filter to cut its intensity in half?

Polarization The angle between the axes of two polarizing filters is \(45.0^{\circ}\). By how much does the second filter reduce the intensity of the light coming through the first? Text Transcription: 45.0 degrees

Polarization Two polarizing sheets \(P_{1} \text { and } P_{2}\) are placed together with their transmission axes oriented at an angle \(\theta\) to each other. What is \(\theta\) when only 25% of the maximum transmitted light intensity passes through them? Text Transcription: P_1 and P_2 theta

Polarization Suppose that in the preceding problem the light incident on \(P_{1}\) is unpolarized. At the determined value of \(\theta\), what fraction of the incident light passes through the combination? Text Transcription: P_1 theta

Polarization If you have completely polarized light of intensity \(150 \mathrm{~W} / \mathrm{m}^{2}\) , what will its intensity be after passing through a polarizing filter with its axis at an \(89.0^{\circ}\) angle to the light’s polarization direction? Text Transcription: 150 W/m^2 89.0 degrees

Polarization What angle would the axis of a polarizing filter need to make with the direction of polarized light of intensity \(1.00 \mathrm{~kW} / \mathrm{m}^{2}\) to reduce the intensity to \(10.0 \mathrm{~W} / \mathrm{m}^{2}\)? Text Transcription: 1.00 kW/m^2 10.0 W/m^2

Polarization At the end of Example 1.7 , it was stated that the intensity of polarized light is reduced to 90% of its original value by passing through a polarizing filter with its axis at an angle of \(18.4^{\circ}\) to the direction of polarization. Verify this statement. Text Transcription: 18.4 degrees

Polarization Show that if you have three polarizing filters, with the second at an angle of \(45.0^{\circ}\) to the first and the third at an angle of \(90.0^{\circ}\) to the first, the intensity of light passed by the first will be reduced to 25.0% of its value. (This is in contrast to having only the first and third, which reduces the intensity to zero, so that placing the second between them increases the intensity of the transmitted light.) Text Transcription: 45.0 degrees 90.0 degrees

Polarization Three polarizing sheets are placed together such that the transmission axis of the second sheet is oriented at \(25.0^{\circ}\) to the axis of the first, whereas the transmission axis of the third sheet is oriented at \(40.0^{\circ}\) (in the same sense) to the axis of the first. What fraction of the intensity of an incident unpolarized beam is transmitted by the combination? Text Transcription: 25.0 degrees 40.0 degrees

Polarization In order to rotate the polarization axis of a beam of linearly polarized light by \(90.0^{\circ}\), a student places sheets \(P_{1} \text { and } P_{2}\) with their transmission axes at \(45.0^{\circ}\) and \(90.0^{\circ}\), respectively, to the beam’s axis of polarization. (a) What fraction of the incident light passes through \(P_{1}\) and (b) through the combination? (c) Repeat your calculations for part (b) for transmission-axis angles of \(30.0^{\circ}\) and \(90.0^{\circ}\), respectively. Text Transcription: 90.0 degrees P_1 and P_2 45.0 degrees P_1 30.0 degrees

Polarization It is found that when light traveling in water falls on a plastic block, Brewster’s angle is \(50.0^{\circ}\).What is the refractive index of the plastic? Text Transcription: 50.0 degrees

Polarization At what angle will light reflected from diamond be completely polarized?

Polarization What is Brewster’s angle for light traveling in water that is reflected from crown glass?

Polarization A scuba diver sees light reflected from the water’s surface. At what angle relative to the water’s surface will this light be completely polarized?

(a) Light reflected at \(62.5^{\circ}\) from a gemstone in a ring is completely polarized. Can the gem be a diamond? (b) At what angle would the light be completely polarized if the gem was in water? Text Transcription: 62.5 degrees

If \(\theta_{\mathrm{b}}\) is Brewster’s angle for light reflected from the top of an interface between two substances, and \(\theta_{\mathrm{b}}^{\prime}\) is Brewster’s angle for light reflected from below, prove that \(\theta_{\mathrm{b}}+\theta_{\mathrm{b}}^{\prime}=90.0^{\circ}\). Text Transcription: theta_b theta_b’ Theta_b+theta b’=90.0 degrees

Unreasonable results Suppose light travels from water to another substance, with an angle of incidence of \(10.0^{\circ}\) and an angle of refraction of \(14.9^{\circ}\). (a) What is the index of refraction of the other substance? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent? Text Transcription: 10.0 degrees 14.9 degrees

Unreasonable results Light traveling from water to a gemstone strikes the surface at an angle of \(80.0^{\circ}\) and has an angle of refraction of \(15.2^{\circ}\). (a) What is the speed of light in the gemstone? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent? Text Transcription: 80.0 degrees 15.2 degrees

If a polarizing filter reduces the intensity of polarized light to 50.0% of its original value, by how much are the electric and magnetic fields reduced?

Suppose you put on two pairs of polarizing sunglasses with their axes at an angle of \(15.0^{\circ}\). How much longer will it take the light to deposit a given amount of energy in your eye compared with a single pair of sunglasses? Assume the lenses are clear except for their polarizing characteristics. Text Transcription: 15.0 degrees

If the apex angle \(\phi\) in the previous problem is \(20.0^{\circ} \text { and } n=1.50\), what is the value of \(\alpha\). Text Transcription: phi 20.0 degrees and n=1.50 alpha

The light incident on polarizing sheet \(P_{1}\) is linearly polarized at an angle of \(30.0^{\circ}\) with respect to the transmission axis of \(P_{1}\). Sheet \(P_{2}\) is placed so that its axis is parallel to the polarization axis of the incident light, that is, also at \(30.0^{\circ}\) with respect to \(P_{1}\). (a) What fraction of the incident light passes through \(P_{1}\)? (b) What fraction of the incident light is passed by the combination? (c) By rotating \(P_{2}\), a maximum in transmitted intensity is obtained. What is the ratio of this maximum intensity to the intensity of transmitted light when \(P_{2}\) is at \(30.0^{\circ}\) with respect to \(P_{1}\)? Text Transcription: P_1 30 degrees P_2

Prove that if I is the intensity of light transmitted by two polarizing filters with axes at an angle \(\theta\) and I’ is the intensity when the axes are at an angle \(90.0^{\circ}-\theta, \text { then } I+I^{\prime}=I_{0}\), the original intensity. (Hint: Use the trigonometric identities \(\cos 90.0^{\circ}-\theta=\sin \theta \text { and } \cos ^{2} \theta+\sin ^{2} \theta=1\).) Text Transcription: theta 90.0 degrees - theta, then I + I' = I_0 cos 90.0 degrees - theta = sin theta and cos^2 theta + sin^2 theta=1

(a) On a day when the intensity of sunlight is \(1.00 \mathrm{~kW} / \mathrm{m}^{2}\), a circular lens 0.200m in diameter focuses light onto water in a black beaker. Two polarizing sheets of plastic are placed in front of the lens with their axes at an angle of \(20.0^{\circ}\). Assuming the sunlight is unpolarized and the polarizers are 100% efficient, what is the initial rate of heating of the water in \({ }^{\circ} \mathrm{C} / \mathrm{s}\), assuming it is 80.0% absorbed? The aluminum beaker has a mass of 30.0 grams and contains 250 grams of water (b) Do the polarizing filters get hot? Explain. Text Transcription: 1.00 kW/m^2 20.0 degrees F degree C/s